Quotients of some identifying subcategories of the topological category

In this paper, let $\mathbf{K}$ denote an identifying subcategory of the category of all topological spaces such that the front adjunction mapping is a quotient mapping for any topological space. The objects in $\mathbf{K}$ are called $\mathbf{K}$-spaces. Note that $\mathbf{K}$ is reflective and $\mathbf{Top}_{i}$ ($i=0$, $1$ and $2$) is a special case for $\mathbf{K}$. For a $\mathbf{K}$-space $X$, we give sufficient and necessary conditions for $X/R$ to be a $\mathbf{K}$-space. Meanwhile, we get the Isomorphism Theorem for $\mathbf{K}$-spaces, with a particular emphasis on $T_{i}$ spaces. For a $T_{i}$ space $(X,\tau)$, we obtain the sufficient and necessary conditions for $X/R$ to be $T_{i}$ only in terms of the given equivalence relation and the space itself. Especially, when $X$ is $T_{2}$, we find that $X/R$ is $T_{2}$ if and only if $R\subseteq (X,\tau_{R})\times (X,\tau_{R})$ is closed. Moreover, we obtain the congruence relation of posets from the topological perspective.